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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 | import Mathlib
import OAI.Geometry.Borsuk.Support.ActiveChambers
import OAI.Geometry.Borsuk.Support.NegativeFibre
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator Topology
open Set
section PurityChart
variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι]
local notation "E" => EuclideanSpace ℝ ν
local notation "M" => Matrix ν ν ℝ
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
noncomputable def purityChart (f : E → ι → ℝ) (I : Finset ι) (w c : E) : Set M :=
{A | IsUnit A ∧ HasPositiveLine A ∧ op A⁺ w ≠ 0 ∧ 0 < Matrix.trace A⁻ ∧
lineRepresentative w A ∈ commonPositiveCone f I ∧
lineRepresentative w A ∈ interior (signChamber (activeVectors f I) c)}
omit [Nonempty ν] in
lemma Admissible.purityChart_isOpen {f : E → ι → ℝ} (hf : Admissible f)
(I : Finset ι) (w c : E) {X : Type*} [TopologicalSpace X]
(A : X → M) (hA : Continuous A) (hSA : ∀ x, IsSelfAdjoint (A x)) :
IsOpen (A ⁻¹' purityChart f I w c) := by
have hop : Continuous (fun x => op (A x)⁺ w) :=
(((op).toAlgEquiv.toLinearEquiv.toLinearMap.continuous_of_finiteDimensional).comp
(matrix_posPart_continuousOn.comp_continuous hA hSA)).clm_apply continuous_const
have hopen : IsOpen {x | op (A x)⁺ w ≠ 0} := (isClosed_eq hop continuous_const).isOpen_compl
have hrep := lineRepresentative_continuousOn A hA hSA w
have hpre : IsOpen ({x | op (A x)⁺ w ≠ 0} ∩
(fun x => lineRepresentative w (A x)) ⁻¹'
(commonPositiveCone f I ∩ interior (signChamber (activeVectors f I) c))) :=
hrep.isOpen_inter_preimage hopen ((hf.commonPositiveCone_isOpen I).inter isOpen_interior)
have hn : IsOpen {x | 0 < Matrix.trace (A x)⁻} := isOpen_lt continuous_const
(matrix_trace_continuous.comp (matrix_negPart_continuousOn.comp_continuous hA hSA))
have he : A ⁻¹' purityChart f I w c =
({x | IsUnit (A x) ∧ HasPositiveLine (A x)} ∩ {x | 0 < Matrix.trace (A x)⁻}) ∩
({x | op (A x)⁺ w ≠ 0} ∩ (fun x => lineRepresentative w (A x)) ⁻¹'
(commonPositiveCone f I ∩ interior (signChamber (activeVectors f I) c))) := by
ext x
simp only [purityChart,mem_preimage,mem_ofPred_eq,mem_inter_iff]
tauto
rw [he]
exact ((positive_line_isOpen A hA hSA).inter hn).inter hpre
omit [Fintype ι] in
omit [Nonempty ν] in
lemma purityChart_unit {f : E → ι → ℝ} {I : Finset ι} {w c : E} {A : M}
(hA : A ∈ purityChart f I w c) : ‖lineRepresentative w A‖ = 1 :=
lineRepresentative_norm hA.2.2.1
omit [Fintype ι] in
lemma purityChart_fibre {f : E → ι → ℝ} {I : Finset ι} {w c : E} {A : M}
(hA : A ∈ purityChart f I w c) : negativeFibre A ∈ matrixFibre (lineRepresentative w A) :=
negativeFibre_mem hA.2.1 hA.2.2.1 hA.2.2.2.1
omit [Fintype ι] in
lemma purityChart_fibre_unit {f : E → ι → ℝ} {I : Finset ι} {w c : E} {A : M}
(hSA : IsSelfAdjoint A) (hA : A ∈ purityChart f I w c) :
IsUnit (rankOne (lineRepresentative w A) + negativeFibre A) := by
have hf := purityChart_fibre hA
apply fibre_unit_of_split_unit hf.1 hf.2.2 (hasPositiveLine_trace_pos hA.2.1) hA.2.2.2.1
rw [← negativeFibre_splitting hSA hA.2.1 hA.2.2.1 hA.2.2.2.1]
exact hA.1
end PurityChart
end BorsukNine
end OAI
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