| import Mathlib |
| import OAI.Geometry.Borsuk.Simplicial.LabelRegion |
| |
| namespace OAI |
|
|
| namespace BorsukNine.SimplicialChains |
| open Finset |
|
|
| noncomputable def signVector (m : ℕ) (b : Fin m → Bool) : EuclideanSpace ℝ (Fin m) := |
| WithLp.toLp 2 (fun i => if b i then (-1:ℝ) else 1) |
|
|
| @[simp] lemma signVector_apply (m : ℕ) (b : Fin m → Bool) (i : Fin m) : |
| signVector m b i = if b i then (-1:ℝ) else 1 := rfl |
|
|
| lemma signVector_norm (m : ℕ) (b : Fin m → Bool) : ‖signVector m b‖ = Real.sqrt m := by |
| rw [EuclideanSpace.norm_eq] |
| congr 1 |
| have h (i : Fin m) : ‖signVector m b i‖ ^ 2 = (1:ℝ) := by |
| simp only [signVector_apply] |
| cases b i < |
| simp only [h, Finset.sum_const, nsmul_eq_mul, Finset.card_univ, Fintype.card_fin, mul_one] |
| |
| noncomputable def facetCenter (m : ℕ) (b : Fin m → Bool) : EuclideanSpace ℝ (Fin m) := |
| (Real.sqrt m)⁻¹ • signVector m b |
| |
| lemma facetCenter_norm (m : ℕ) [NeZero m] (b : Fin m → Bool) : ‖facetCenter m b‖ = 1 := by |
| rw [facetCenter, norm_smul, signVector_norm, Real.norm_eq_abs, abs_of_nonneg (inv_nonneg.mpr (Real.sqrt_nonneg _))] |
| exact inv_mul_cancel₀ (ne_of_gt (Real.sqrt_pos_of_pos (by exact_mod_cast NeZero.pos m))) |
|
|
| lemma facetCenter_region (m : ℕ) (b : Fin m → Bool) (i : Fin m) : |
| facetCenter m b ∈ labelRegion i (b i) := by |
| have ha (j : Fin m) : |facetCenter m b j| = (Real.sqrt m)⁻¹ := by |
| simp only [facetCenter, PiLp.smul_apply, smul_eq_mul, signVector_apply, abs_mul, |
| abs_of_nonneg (inv_nonneg.mpr (Real.sqrt_nonneg _))] |
| cases b j < |
| refine ⟨fun j => by rw [ha j,ha i],?_⟩ |
| change (if b i then (Real.sqrt m)⁻¹ * (if b i then (-1:ℝ) else 1) ≤ 0 |
| else 0 ≤ (Real.sqrt m)⁻¹ * (if b i then (-1:ℝ) else 1)) |
| have h := inv_nonneg.mpr (Real.sqrt_nonneg (m:ℝ)) |
| cases b i < |
|
|
| lemma all_labelRegions_iff_center (m : ℕ) [NeZero m] (b : Fin m → Bool) |
| (y : EuclideanSpace ℝ (Fin m)) (hy : ‖y‖ = 1) : |
| (∀ i, y ∈ labelRegion i (b i)) ↔ y = facetCenter m b := by |
| refine ⟨?_,fun he i => he ▸ facetCenter_region m b i⟩ |
| intro hyc |
| let i₀ : Fin m := ⟨0,NeZero.pos m⟩ |
| let d : ℝ := |y i₀| |
| have hd : 0 ≤ d := abs_nonneg _ |
| have hcoord (i : Fin m) : |y i| = d := |
| le_antisymm ((hyc i₀).1 i) ((hyc i).1 i₀) |
| have hv : y = d • signVector m b := by |
| ext i |
| simp only [PiLp.smul_apply, smul_eq_mul, signVector_apply] |
| have hsign := (hyc i).2 |
| change (if b i then y i ≤ 0 else 0 ≤ y i) at hsign |
| cases hb : b i |
| · simp only [hb, Bool.false_eq_true, ↓reduceIte] at hsign ⊢ |
| simpa only [abs_of_nonneg hsign, mul_one] using hcoord i |
| · simp only [hb, ↓reduceIte] at hsign ⊢ |
| have he := hcoord i |
| rw [abs_of_nonpos hsign] at he |
| linarith |
| have hdval : d = (Real.sqrt m)⁻¹ := by |
| have he := hy |
| rw [hv,norm_smul,Real.norm_eq_abs,abs_of_nonneg hd,signVector_norm] at he |
| exact eq_inv_of_mul_eq_one_left he |
| rw [hv,hdval] |
| rfl |
|
|
| lemma constant_approximation_facet_zero (n m r : ℕ) (hm : 1 < m) [NeZero m] |
| (b : Fin m → Bool) (y : EuclideanSpace ℝ (Fin m)) : |
| approximation n m r (fun _ => sphereLabel y) (crossFundamental n) |
| ⟨crossFacet m b,crossFacet_mem m (NeZero.pos m) b⟩ = 0 := by |
| classical |
| let K := hemisphereComplex (n:ℝ) (E := EuclideanSpace ℝ (Fin n)) |
| let t : {s : Finset SignedVertex // s ∈ cross m} := ⟨crossFacet m b,crossFacet_mem m (NeZero.pos m) b⟩ |
| let T : Chain K →ₗ[F₂] F₂ := (Finsupp.lapply t).comp (mapChains K (cross m) (fun _ => sphereLabel y)) |
| change T (fineCrossMap n r (crossFundamental n)) = 0 |
| have ht : T = 0 := by |
| apply Finsupp.lhom_ext |
| intro s a |
| have hh : T (simplex K s.val) = 0 := by |
| change mapChains K (cross m) (fun _ => sphereLabel y) (simplex K s.val) t = 0 |
| rw [mapChains_simplex K (cross m) _ s.property] |
| apply mapSimplex_coeff_zero_of_ne |
| intro he |
| have hc : (s.val.image (fun _ => sphereLabel y)).card ≤ 1 := by |
| apply Finset.card_le_one.mpr |
| intro a ha c hc |
| obtain ⟨_,_,rfl⟩ := Finset.mem_image.mp ha |
| obtain ⟨_,_,rfl⟩ := Finset.mem_image.mp hc |
| rfl |
| rw [he,crossFacet_card] at hc |
| omega |
| rw [simplex_of_mem K s.property] at hh |
| have ha : Finsupp.single s a = a • Finsupp.single s (1:F₂) := by simp |
| simp only [ha,map_smul,hh,smul_zero,LinearMap.zero_apply] |
| rw [ht,LinearMap.zero_apply] |
|
|
| lemma continuous_missed_facet_zero (n m : ℕ) (hn : 0 < n) (hm : 1 < m) [NeZero m] |
| (b : Fin m → Bool) (f : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m)) |
| (hf : ContinuousOn f (annulus n)) (hfn : ∀ x ∈ annulus n, ‖f x‖ = 1) |
| (hmiss : ∀ x ∈ annulus n, f x ≠ facetCenter m b) : |
| ∃ δ > 0, ∀ r : ℕ, (((n:ℝ)-1)/n)^r * 2 < δ → |
| approximation n m r (fun x => sphereLabel (f x)) (crossFundamental n) |
| ⟨crossFacet m b,crossFacet_mem m (NeZero.pos m) b⟩ = 0 := by |
| have hfneg : -(facetCenter m b) ≠ facetCenter m b := by |
| intro he |
| have hz : facetCenter m b = 0 := by |
| ext i |
| have hi := congrArg (fun z : EuclideanSpace ℝ (Fin m) => z i) he |
| simp only [PiLp.neg_apply] at hi |
| change facetCenter m b i = 0 |
| linarith |
| have hn := facetCenter_norm m b |
| rw [hz,norm_zero] at hn |
| exact zero_ne_one hn |
| obtain ⟨δ,hδ,hh⟩ := localized_approximation_coeff n m hn b f (fun _ => -(facetCenter m b)) |
| hf continuousOn_const ∅ isOpen_empty (by simp) (by |
| intro x hx |
| rcases hx with h | h |
| · exact False.elim (hmiss x.val x.property ((all_labelRegions_iff_center m b _ (hfn _ x.property)).mp h)) |
| · exact False.elim (hfneg ((all_labelRegions_iff_center m b _ (by simp [facetCenter_norm])).mp h))) |
| refine ⟨δ,hδ,fun r hr => ?_⟩ |
| rw [hh r hr,constant_approximation_facet_zero n m r hm b] |
| |
| end BorsukNine.SimplicialChains |
| |
| end OAI |
| |
