File size: 3,136 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 | 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
end OAI
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