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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 | import Mathlib
import OAI.Geometry.Borsuk.MatrixGeometry.MatrixCompression
import OAI.Geometry.Borsuk.MatrixGeometry.SymmetricEntries
import OAI.Geometry.Borsuk.MatrixGeometry.FibreContinuity
import OAI.Geometry.Borsuk.Projectors.CoordinateSum
import OAI.Geometry.Borsuk.Projectors.FiniteSmoothSliceMissesBounded
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
open Set
section CenterApproximation
variable {k : ℕ} {ι : Type*} [Fintype ι] [DecidableEq ι]
local notation "D" => EuclideanSpace ℝ (UpperTriangle (Fin k))
local notation "W" => EuclideanSpace ℝ (Fin k)
local notation "E" => EuclideanSpace ℝ (Fin (k+1))
local notation "N" => Matrix (Fin k) (Fin k) ℝ
local notation "M" => Matrix (Fin (k+1)) (Fin (k+1)) ℝ
local notation "op" => Matrix.toEuclideanCLM (n := Fin (k+1)) (𝕜 := ℝ)
lemma matrixFibre_range {u : E} (e : W →ₗᵢ[ℝ] E)
(he : ∀ v, ⟪u,v⟫ = 0 → ∃ x, e x = v) {Q : M} (hQ : Q ∈ matrixFibre u) :
LinearMap.range (Matrix.toEuclideanLin Q) ≤ LinearMap.range e.toLinearMap := by
rintro v ⟨b,rfl⟩
obtain ⟨x,hx⟩ := he (op Q b) (matrix_image_orthogonal_kernel hQ.1.isSelfAdjoint hQ.2.2 b)
exact ⟨x,hx⟩
lemma matrixCompress_mem_slice {u : E} (e : W →ₗᵢ[ℝ] E)
(he : ∀ v, ⟪u,v⟫ = 0 → ∃ x, e x = v) {Q : M} (hQ : Q ∈ matrixFibre u) :
0 ≤ matrixCompress e Q ∧ Matrix.trace (matrixCompress e Q) = 1 :=
⟨matrixCompress_nonneg e hQ.1,
(matrixCompress_trace e hQ.1.isSelfAdjoint (matrixFibre_range e he hQ)).trans hQ.2.1⟩
lemma Admissible.center_approximation {f : E → ι → ℝ} (hf : Admissible f)
{I : Finset ι} [Nonempty ↥(Iᶜ : Finset ι)] {c : E}
(hc : c ∈ safeNormals (activeVectors f I)) (e : W →ₗᵢ[ℝ] E)
(he : ∀ x, ⟪c,e x⟫ = 0) {ε : ℝ} (hε : 0 < ε) :
∃ a : D → EuclideanSpace ℝ ↥(Iᶜ : Finset ι), ContDiff ℝ (⊤ : ℕ∞) a ∧
(∀ x, ∑ i, a x i = 1) ∧
∀ x, 0 ≤ symmetricCoordinates x → Matrix.trace (symmetricCoordinates x) = 1 →
‖a x - negativeCoordinates f I (matrixEmbed e (symmetricCoordinates x))‖ < ε := by
let g : D → EuclideanSpace ℝ ↥(Iᶜ : Finset ι) :=
fun x => negativeCoordinates f I (matrixEmbed e ((symmetricCoordinates x)⁺))
have hP : Continuous (fun x : D => (symmetricCoordinates x)⁺) :=
matrix_posPart_continuousOn.comp_continuous symmetricCoordinates.continuous_of_finiteDimensional
(fun x => symmetricCoordinates_selfAdjoint x)
have hg : Continuous g :=
(hf.negativeCoordinates_continuousOn I).comp_continuous
((matrixEmbed e).continuous_of_finiteDimensional.comp hP)
(fun x => matrixEmbed_nonneg e (CFC.posPart_nonneg _))
let K : Set D := {x | 0 ≤ symmetricCoordinates x ∧ Matrix.trace (symmetricCoordinates x) = 1}
have hs : ∀ x ∈ K, coordinateSum (g x) = 1 := by
intro x hx
change ∑ i, negativeCoordinates f I (matrixEmbed e ((symmetricCoordinates x)⁺)) i = 1
rw [(CFC.posPart_eq_self _).mpr hx.1]
apply hf.negativeCoordinates_sum hc
exact ⟨matrixEmbed_nonneg _ hx.1,(matrixEmbed_trace _ _).trans hx.2,matrixEmbed_kernel e _ he⟩
obtain ⟨a,ha,has,hae⟩ := continuous_simplex_approx g hg hs hε
refine ⟨a,ha,has,?_⟩
intro x hx ht
have hh := hae x ⟨hx,ht⟩
simpa only [g,(CFC.posPart_eq_self _).mpr hx] using hh
lemma Admissible.finite_center_approximations {κ : Type*} [Fintype κ]
{f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
(hd : Fintype.card (UpperTriangle (Fin k)) < Fintype.card ↥(Iᶜ : Finset ι))
(c : κ → E) (hc : ∀ j, c j ∈ safeNormals (activeVectors f I)) (hcn : ∀ j, ‖c j‖ = 1) :
∃ z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι), (∀ i, (2*(Fintype.card ↥(Iᶜ : Finset ι):ℝ))⁻¹ < z i) ∧
(∑ i, z i) = 1 ∧
∃ g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι), (∀ j, Continuous (g j)) ∧
(∀ j, ∀ Q ∈ matrixFibre (c j), (∑ i, g j Q i) = 1) ∧
(∀ j, ∀ Q ∈ matrixFibre (c j), ‖g j Q - negativeCoordinates f I Q‖ <
(2*(Fintype.card ↥(Iᶜ : Finset ι):ℝ))⁻¹) ∧
(∀ j, ∀ Q ∈ matrixFibre (c j), g j Q ≠ z) := by
have hj : 0 < Fintype.card ↥(Iᶜ : Finset ι) := lt_of_le_of_lt (Nat.zero_le _) hd
let : Nonempty ↥(Iᶜ : Finset ι) := Fintype.card_pos_iff.mp hj
let ε : ℝ := (2*(Fintype.card ↥(Iᶜ : Finset ι):ℝ))⁻¹
have hε : 0 < ε := by dsimp [ε]; positivity
choose e he hes using (fun j => orthogonal_frame k (hcn j))
choose a ha has hae using (fun j => hf.center_approximation (hc j) (e j) (he j) hε)
let L : D →L[ℝ] ℝ := (Matrix.traceLinearMap (Fin k) ℝ ℝ).toContinuousLinearMap.comp
(symmetricCoordinates (ν := Fin k)).toContinuousLinearMap
have hendim : Module.finrank ℝ D < Fintype.card ↥(Iᶜ : Finset ι) := by simpa using hd
obtain ⟨z,hzp,hzs,hzm⟩ := finite_smooth_slice_misses_general hendim (fun _ : κ => L) a ha
let g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι) :=
fun j Q => a j (symmetricEntries (matrixCompress (e j) Q))
refine ⟨z,hzp,hzs,g,?_,?_,?_,?_⟩
· intro j
exact (ha j).continuous.comp (symmetricEntries.continuous_of_finiteDimensional.comp
(matrixCompress (e j)).continuous_of_finiteDimensional)
· intro j Q hQ
exact has j _
· intro j Q hQ
have hCQ := matrixCompress_mem_slice (e j) (hes j) hQ
have hSC := symmetricCoordinates_entries hCQ.1.isSelfAdjoint
have hEQ := matrixEmbed_compress (e j) hQ.1.isSelfAdjoint (matrixFibre_range (e j) (hes j) hQ)
have hh := hae j (symmetricEntries (matrixCompress (e j) Q))
(by rw [hSC]; exact hCQ.1) (by rw [hSC]; exact hCQ.2)
simpa only [hSC,hEQ,g,ε] using hh
· intro j Q hQ
apply hzm j _
change Matrix.trace (symmetricCoordinates (symmetricEntries (matrixCompress (e j) Q))) = 1
have hCQ := matrixCompress_mem_slice (e j) (hes j) hQ
rw [symmetricCoordinates_entries hCQ.1.isSelfAdjoint]
exact hCQ.2
end CenterApproximation
end BorsukNine
end OAI
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