File size: 5,924 Bytes
8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | 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 <;> norm_num
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 <;> norm_num
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 <;> simp_all
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
|