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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import Mathlib
import OAI.Geometry.Borsuk.Simplicial.LabelRegion
namespace OAI
namespace BorsukNine.SimplicialChains
open Finset
lemma compact_additive_locality {X A : Type*} [MetricSpace X] [CompactSpace X]
[Fintype A] [DecidableEq A] {m : ℕ} [NeZero m]
(b : Fin m → Bool) (f : X → EuclideanSpace ℝ (Fin m))
(g : A → X → EuclideanSpace ℝ (Fin m)) (hf : Continuous f) (hg : ∀ a, Continuous (g a))
(U : A → Set X) (hU : ∀ a, IsOpen (U a))
(heq : ∀ a x, x ∈ U a → f x = g a x)
(hcover : ∀ x, (∀ i, f x ∈ labelRegion i (b i)) → ∃ a, x ∈ U a)
(hroot : ∀ a x, (∀ i, g a x ∈ labelRegion i (b i)) → x ∈ U a)
(hdis : Pairwise (fun a a' => Disjoint (U a) (U a'))) :
∃ δ > 0, ∀ x : X,
(∃ i : Fin m, (∀ y, dist y x < δ → sphereLabel (f y) ≠ (i.val,b i)) ∧
∀ a, ∃ j : Fin m, ∀ y, dist y x < δ → sphereLabel (g a y) ≠ (j.val,b j)) ∨
∃ a, (∀ y, dist y x < δ → f y = g a y) ∧
∀ a' ≠ a, ∃ j : Fin m, ∀ y, dist y x < δ → sphereLabel (g a' y) ≠ (j.val,b j) := by
classical
let c : Option A × (Option A → Fin m) → Set X := fun p => match p.1 with
| none => (f ⁻¹' labelRegion (p.2 none) (b (p.2 none)))ᶜ ∩
⋂ a, (g a ⁻¹' labelRegion (p.2 (some a)) (b (p.2 (some a))))ᶜ
| some a => U a ∩ ⋂ a' : {a' : A // a' ≠ a},
(g a'.val ⁻¹' labelRegion (p.2 (some a'.val)) (b (p.2 (some a'.val))))ᶜ
have hopen : ∀ p, IsOpen (c p) := by
intro p
cases ho : p.1 with
| none =>
simp only [c,ho]
exact (((labelRegion_closed _ _).preimage hf).isOpen_compl).inter
(isOpen_iInter_of_finite (fun a => ((labelRegion_closed _ _).preimage (hg a)).isOpen_compl))
| some a =>
simp only [c,ho]
exact (hU a).inter (isOpen_iInter_of_finite
(fun a' => ((labelRegion_closed _ _).preimage (hg a'.val)).isOpen_compl))
have hc : (Set.univ : Set X) ⊆ ⋃ p, c p := by
intro x _
by_cases hx : ∃ a, x ∈ U a
· obtain ⟨a,ha⟩ := hx
have hm : ∀ a' : {a' : A // a' ≠ a}, ∃ j, g a'.val x ∉ labelRegion j (b j) := by
intro a'
apply not_forall.mp
intro hh
exact Set.disjoint_left.mp (hdis a'.property) (hroot a'.val x hh) ha
choose j hj using hm
let ji : Option A → Fin m := fun o => match o with
| none => ⟨0,NeZero.pos m⟩
| some a' => if h : a' ≠ a then j ⟨a',h⟩ else ⟨0,NeZero.pos m⟩
refine Set.mem_iUnion.mpr ⟨(some a,ji),ha,Set.mem_iInter.mpr ?_⟩
intro a'
simpa only [ji,dite_eq_left a'.property,Set.mem_compl_iff,Set.mem_preimage] using hj a'
· have hm : ∀ o : Option A, ∃ j, (match o with | none => f x | some a => g a x) ∉ labelRegion j (b j) := by
intro o
apply not_forall.mp
intro hh
cases o with
| none => exact hx (hcover x hh)
| some a => exact hx ⟨a,hroot a x hh⟩
choose j hj using hm
exact Set.mem_iUnion.mpr ⟨(none,j),hj none,Set.mem_iInter.mpr (fun a => hj (some a))⟩
obtain ⟨δ,hδ,hball⟩ := lebesgue_number_lemma_of_metric isCompact_univ hopen hc
refine ⟨δ,hδ,?_⟩
intro x
obtain ⟨⟨o,j⟩,hj⟩ := hball x (Set.mem_univ _)
cases o with
| none =>
refine Or.inl ⟨j none,?_,?_⟩
· intro y hy he
exact (hj hy).1 (sphereLabel_mem_region _ _ _ he)
· intro a
refine ⟨j (some a),?_⟩
intro y hy he
exact Set.mem_iInter.mp (hj hy).2 a (sphereLabel_mem_region _ _ _ he)
| some a =>
refine Or.inr ⟨a,fun y hy => heq a y (hj hy).1,?_⟩
intro a' ha'
refine ⟨j (some a'),?_⟩
intro y hy he
exact Set.mem_iInter.mp (hj hy).2 ⟨a',ha'⟩ (sphereLabel_mem_region _ _ _ he)
lemma additive_approximation_coeff {A : Type*} [Fintype A] [DecidableEq A]
(n m : ℕ) (hn : 0 < n) [NeZero m] (b : Fin m → Bool)
(f : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m))
(g : A → EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m))
(hf : ContinuousOn f (annulus n)) (hg : ∀ a, ContinuousOn (g a) (annulus n))
(U : A → Set (annulus n)) (hU : ∀ a, IsOpen (U a))
(heq : ∀ a (x : annulus n), x ∈ U a → f x.val = g a x.val)
(hcover : ∀ x : annulus n, (∀ i, f x.val ∈ labelRegion i (b i)) → ∃ a, x ∈ U a)
(hroot : ∀ a (x : annulus n), (∀ i, g a x.val ∈ labelRegion i (b i)) → x ∈ U a)
(hdis : Pairwise (fun a a' => Disjoint (U a) (U a'))) :
∃ 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⟩ =
∑ a, approximation n m r (fun x => sphereLabel (g a x)) (crossFundamental n)
⟨crossFacet m b,crossFacet_mem m (NeZero.pos m) b⟩ := by
classical
let : CompactSpace (annulus n) := isCompact_iff_compactSpace.mp (annulus_compact n)
obtain ⟨δ,hδ,hlocal⟩ := compact_additive_locality b (fun x : annulus n => f x.val)
(fun a (x : annulus n) => g a x.val) hf.domRestrict (fun a => (hg a).domRestrict)
U hU heq hcover hroot hdis
refine ⟨δ,hδ,?_⟩
intro r hr
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 x => sphereLabel (f x)))
let R : A → Chain K →ₗ[F₂] F₂ := fun a =>
(Finsupp.lapply t).comp (mapChains K (cross m) (fun x => sphereLabel (g a x)))
change T (fineCrossMap n r (crossFundamental n)) = ∑ a, R a (fineCrossMap n r (crossFundamental n))
rw [← LinearMap.sum_apply]
apply linearMap_eq_of_support T (∑ a, R a) _
intro s hs
rw [LinearMap.sum_apply]
have heT : T (simplex K s.val) = mapSimplex (cross m) (fun x => sphereLabel (f x)) s.val t := by
exact congrArg (fun c : Chain (cross m) => c t) (mapChains_simplex K (cross m) _ s.property)
have heR (a : A) : R a (simplex K s.val) = mapSimplex (cross m) (fun x => sphereLabel (g a x)) s.val t := by
exact congrArg (fun c : Chain (cross m) => c t) (mapChains_simplex K (cross m) _ s.property)
simp only [heT,heR]
have ha : ∀ x ∈ s.val, x ∈ annulus n := fun x hx => hemisphere_vertex_annulus n hn s.property hx
obtain ⟨x,hx⟩ := s.property.1
have hmesh := fineCrossMap_mesh n r (crossFundamental n) hs
have hball (y) (hy : y ∈ s.val) : dist (⟨y,ha y hy⟩ : annulus n) ⟨x,ha x hx⟩ < δ :=
(hmesh y hy x hx).trans_lt hr
rcases hlocal ⟨x,ha x hx⟩ with ⟨i,hi,hgmiss⟩ | ⟨a,heq,hgmiss⟩
· rw [mapSimplex_coeff_zero_of_ne (cross m) _ s.val t
(image_ne_crossFacet_of_missing s.val _ b i (fun y hy => hi ⟨y,ha y hy⟩ (hball y hy)))]
symm
apply Finset.sum_eq_zero
intro a _
obtain ⟨j,hj⟩ := hgmiss a
exact mapSimplex_coeff_zero_of_ne (cross m) _ s.val t
(image_ne_crossFacet_of_missing s.val _ b j (fun y hy => hj ⟨y,ha y hy⟩ (hball y hy)))
· rw [Finset.sum_eq_single a]
· have himg : s.val.image (fun y => sphereLabel (f y)) = s.val.image (fun y => sphereLabel (g a y)) := by
apply Finset.image_congr
intro y hy
exact congrArg sphereLabel (heq ⟨y,ha y hy⟩ (hball y hy))
simp only [mapSimplex,himg]
· intro a' _ ha'
obtain ⟨j,hj⟩ := hgmiss a' ha'
exact mapSimplex_coeff_zero_of_ne (cross m) _ s.val t
(image_ne_crossFacet_of_missing s.val _ b j (fun y hy => hj ⟨y,ha y hy⟩ (hball y hy)))
· simp
end BorsukNine.SimplicialChains
end OAI
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