| 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 |
|
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| end OAI |
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