| import Mathlib |
| import OAI.Geometry.Borsuk.LocalParity.TraceLocalModels |
| import OAI.Geometry.Borsuk.Projectors.FinitePatch |
| import OAI.Geometry.Borsuk.Projectors.FiniteClosedCutoffs |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| namespace BorsukNine |
| open Set SimplicialChains |
| open scoped MatrixOrder Matrix.Norms.L2Operator |
| section TracePatchModels |
| variable {ν ι κ : Type*} [Fintype ν] [LinearOrder ν] [Nonempty ν] [Fintype ι] |
| [Fintype κ] [DecidableEq κ] |
| local notation "Y" => EuclideanSpace ℝ ι |
| omit [Nonempty ν] in |
| lemma trace_patch_models |
| (hc : Fintype.card (UpperTriangle ν) = Fintype.card ι) (hn : 1 < Fintype.card ι) |
| (F : traceSphere ν → Y) (hF : Continuous F) (hFz : ∀ A, F A ≠ 0) |
| (hFo : ∀ A B : traceSphere ν, B.val = -A.val → F B = -F A) |
| (G : κ → traceSphere ν → Y) (hG : ∀ a, Continuous (G a)) |
| (hGz : ∀ a A, G a A ≠ 0) |
| (hGo : ∀ a (A B : traceSphere ν), B.val = -A.val → G a B = -G a A) |
| (K S : κ → Set (traceSphere ν)) (hK : ∀ a, IsClosed (K a)) (hS : ∀ a, IsOpen (S a)) |
| (hKS : ∀ a, K a ⊆ S a) (hKd : Pairwise (fun a b => Disjoint (K a) (K b))) |
| {y : Y} (hy : y ≠ 0) |
| (hcover : ∀ A, (∃ d : ℝ, 0 < d ∧ F A = d • y) → ∃ a, A ∈ K a) |
| (hroot : ∀ a A, (∃ d : ℝ, 0 < d ∧ G a A = d • y) → A ∈ K a) |
| (hseg : ∀ a A, A ∈ S a → ∀ t ∈ Icc (0:ℝ) 1, (1-t) • F A + t • G a A ≠ 0) |
| (hhit : ∀ a A, A ∈ S a → ∀ t ∈ Icc (0:ℝ) 1, |
| (∃ d : ℝ, 0 < d ∧ (1-t) • F A + t • G a A = d • y) → A ∈ K a) : |
| (Fintype.card κ : F₂) = 1 := by |
| classical |
| obtain ⟨U,q,hU,hKU,hqb,hq1,hqS,hqd⟩ := finite_closed_cutoffs K S hK hS hKS hKd |
| let H : ℝ × traceSphere ν → Y := fun p => finitePatch F G (fun a => q a) p.1 p.2 |
| have hH : Continuous H := finitePatch_continuous hF hG (fun a => (q a).continuous) |
| have hHz (t) (ht : t ∈ Icc (0:ℝ) 1) (A) : H (t,A) ≠ 0 := |
| finitePatch_ne_zero F G (fun a => q a) t A hqd ht (fun a => hqb a A) (hFz A) |
| (fun a ha => hseg a A (hqS a A ha)) |
| refine trace_odd_local_models hc hn H hH.continuousOn hHz |
| (fun A B hB => by simpa only [H,finitePatch_zero] using hFo A B hB) |
| G hG hGz hGo U hU ?_ hy ?_ ?_ ?_ |
| · intro a A hA |
| apply finitePatch_one_of_disjoint F G (fun a => q a) A a (hq1 a A hA) |
| intro b hba |
| by_contra hb |
| exact Set.disjoint_left.mp (hqd hba) hb (by |
| change q a A ≠ 0 |
| rw [hq1 a A hA]; exact one_ne_zero) |
| · intro A hA |
| rcases finitePatch_cases F G (fun a => q a) 1 A hqd with he | ⟨a,ha,he⟩ |
| · have hr : ∃ d : ℝ, 0 < d ∧ F A = d • y := by |
| simpa only [H,he] using hA |
| obtain ⟨a,ha⟩ := hcover A hr |
| exact ⟨a,hKU a ha⟩ |
| · refine ⟨a,hKU a (hhit a A (hqS a A ha) (q a A) (hqb a A) ?_)⟩ |
| simpa only [H,he,one_mul] using hA |
| · intro a A hA |
| exact hKU a (hroot a A hA) |
| · intro a b hab |
| exact (hqd hab).mono |
| (fun A hA => by change q a A ≠ 0; rw [hq1 a A hA]; exact one_ne_zero) |
| (fun A hA => by change q b A ≠ 0; rw [hq1 b A hA]; exact one_ne_zero) |
| end TracePatchModels |
| end BorsukNine |
|
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| end OAI |
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